Essentially, your question is about defamation.
Defamation is false communication about a person that tends to hurts the person's reputation.
The communication must be made to other people, not just to the person it's about. It can be spoken or written, or it can even be a gesture.
The law protects your reputation against defamation so, if someone defames you, you can sue the person for money to compensate you for your damage to your reputation.
But it's important to understand that the law doesn't protect you from a personal insult or a remark that injures only your pride; it protects reputation, not feelings.
For example, if someone says that you are a sleazy person then that's not damaging to your reputation as it's really just a mean insult. But if the person is saying that you are dishonest in business, you probably do have a good reason to sue.
In your case because the defamation is happening both verbally and in online the actual legal terms for what you would sue for are "slander" and "libel".
If you can prove that someone libeled you and that person does not have a good defence, then a court will presume that you suffered damages and award you money to compensate for your damaged reputation.
The law treats slander differently than libel: with slander, you have to prove you suffered damages, in the form of financial loss, to get compensation. It can be very difficult to prove this sort of financial loss. That's why most slander cases never go to court.
But in the following four examples, a slander lawsuit may succeed without you proving financial loss. Even though there's no permanent record of the slander, the law will presume damages, as if there were libel, if someone:
- accuses you of a crime (unless they made the accusation to the police)
- accuses you of having a contagious disease
- makes negative remarks about you in your trade or business
- accuses you of adultery
Your next step should be to consult with a lawyer face to face with a view to retaining the lawyer to send this sister a cease and desist letter demanding a formal retraction and written apology and threatening a lawsuit for significant damages.